2 Answers
2

Here's a hint:
$$
\int_{1/e}^1 \frac{1}{\sqrt{\ln x}} {\huge(}\frac{dx}{x}{\huge)}.
$$
If you don't know what that's hinting at, then you don't understand substitutions. It's all about the chain rule. The part in the gigantic parentheses becomes $du$.

I understand, but still my problem is the know when to change the limits, I get : $$\int^1_\frac{1}{e} \frac{dt}{\sqrt{t}}$$
–
Ofir AttiaMay 22 '13 at 17:23

You have $t = \ln x$. When $x = 1/e$, then $t=\ln(1/e)=-1$. When $x=1$, then $t=\ln 1 = 0$. So you have $\displaystyle\int_{-1}^0 \frac{dt}{\sqrt{t}}$. Since you're talking about square roots of negative numbers, you have a question of how to make sense of those. One branch, maybe.
–
Michael HardyMay 22 '13 at 20:54